Summary: For a continuous circle map \(T\), define the barycentre of any \(T\)-invariant probability measure \( \mu\) to be \(b(\mu)=\int_{S^1} z\, d\mu(z)\). The set \(\Omega\) of all such barycentres is a compact convex subset of \(\mathbb{C}\). If \(T\) is conjugate to a rational rotation via a Möbius map, we prove \(\Omega\) is a disc. For every piecewise-onto expanding map we prove that the barycentre set has non-empty interior. In this case, each interior point is the barycentre of many invariant measures, but we prove that amongst these there is a unique one which maximizes entropy, and that this measure belongs to a distinguished two-parameter family of equilibrium states. This family induces a real-analytic radial foliation of \(\text{int}(\Omega)\), centred around the barycentre of the global measure of maximal entropy, where each ray is the barycentre locus of some one-parameter section of the family. We explicitly compute these rays for two examples. While developing this framework we also answer a conjecture of Z. Coelho [Entropy and ergodicity of skew-products over subshifts of finite type and central limit asymptotics. PhD Thesis, Warwick University (1990)] regarding limits of sequences of equilibrium states.